The vectors $$\left( {2\hat i - m\hat j + 3m\hat k} \right)\& \left\{ {\left( {1 + m} \right)\hat i - 2m\hat j + \hat k} \right\}$$         include an acute angle for :

Solution :
$$\overrightarrow a = 2\hat i - m\hat j + 3m\hat k\,\,\& \,\,\overrightarrow b = \left( {1 + m} \right)\hat i - 2m\hat j + \hat k$$
and if angle between $$\overrightarrow a $$ and $$\overrightarrow b $$ is an acute, then $$\overrightarrow a .\overrightarrow b > 0$$
$$\eqalign{ & \Rightarrow 2\left( {1 + m} \right) + 2{m^2} + 3m > 0 \cr & \Rightarrow 2{m^2} + 5m + 2 > 0 \cr & \Rightarrow 2{m^2} + 4m + m + 2 > 0 \cr & \Rightarrow \left( {2m + 1} \right)\left( {m + 2} \right) > 0 \cr & \Rightarrow m < - 2{\text{ or }}m > - \frac{1}{2} \cr} $$